FIG. 1 shows an axial view of a third generation CT scanner 10 that includes an X-ray source 12 and an X-ray detector system 14 secured respectively to diametrically opposite sides of an annular-shaped disk 16. The disk is rotatably mounted within a gantry support (not shown) so that during a scan, the disk continuously rotates about a Z-axis (which is normal to the plane of the page in FIG. 1) while X-rays pass from the source 12 through an object, such as a patient 20, positioned within the opening of the disk to the detector system 14.
The detector system 14 typically includes an array of individual detectors 22 disposed as a single row in the shape of an arc of a circle having a center of curvature at the point 24, referred to as the "focal spot", where the radiation emanates from the X-ray source 12. The X-ray source and the array of detectors are positioned so that the X-ray paths between the source and each detector all lie in a "scanning plane" that is normal to the Z-axis. Since the X-ray paths originate from what is substantially a point source and extend at different angles to the detectors, the X-ray paths form a "fan beam" 26 that is incident on the detector system 14. The X-rays incident on a single detector at a measuring instant during a scan are commonly referred to as a "ray", and each detector generates an output signal indicative of the intensity of its corresponding ray. Since each ray is partially attenuated by all the mass in its path, the output signal generated by each detector is representative of the density of all the mass disposed between that detector and the X-ray source (i.e., the density of the mass lying in the detector's corresponding ray path).
The output signals generated by the X-ray detectors are normally processed by a signal processing portion (not shown) of the CT system. The signal processing portion generally includes a data acquisition system (DAS) which filters the output signals generated by the X-ray detectors to improve their signal-to-noise ratio. One such DAS is described for example in U.S. Pat. No. 4,547,893, which is assigned to the assignee of the present invention, although other systems are known. The filtered output signals generated by the DAS are commonly referred to as "raw data signals". The signal processing portion usually includes a projection filter which logarithmically processes the raw data signals to generate a set of projection data signals so that each projection data signal is representative of the density of the mass lying in a corresponding ray path. The collection of all the projection data signals at a measuring instant or interval is commonly referred to as a "projection" or a "view". The angular orientation of the disk corresponding to a particular projection is referred to as the "projection angle".
FIG. 2 illustrates the orientation of the disk 16 (as well as the X-ray source 12 and detector system 14 mounted to the disk) for generation of a fan beam projection P.sub.f (.beta.) at a projection angle of .beta.. A center line 40, which is used to define reference orientations, extends from the focal spot of the X-ray source 12 through the Z-axis. The projection angle .beta. is defined as the angle between a vertical axis and center line 40. Each fan beam projection P.sub.f (.beta.) may be represented as a set of data points F.sub.1, F.sub.2, . . . , F.sub.N, where the ith data point F.sub.i represents a sample of the projection data signal generated in response to the ith detector of the detector system 14 at a projection angle of .beta., and where the detector system 14 includes N individual detectors.
Each individual detector within detector system 14 has an associated detector angle .gamma. that is defined with respect to center line 40. By definition, center line 40 intersects detector system 14 with a detector angle of .gamma. equal to 0.degree.. A symmetric detector system 14 (as shown in FIG. 2) extends from a detector angle of -.gamma..sub.m to +.gamma..sub.m, and as will be discussed in greater detail below, an asymmetric detector system extends from a detector angle of -.gamma..sub.m +.alpha. to +.gamma..sub.m. Three of the individual detectors in system 14 are indicated at A, B, and C. Detectors A and B are located at opposite ends of the detector system at detector angles of -.gamma..sub.m and +.gamma..sub.m, respectively, and detector C (which is referred to as the central detector) is located at a detector angle of 0.degree.. When the center line 40 intersects the center of the central detector C, the detector system may be described as a "centered" system. Conversely, when the center line 40 intersects the central detector C at a point that is offset from that detector's center, then the detector system may be described as "offset", or as a detector system that is characterized by a detector offset.
During a scan, the disk 16 rotates smoothly and continuously around the object being scanned allowing the scanner 10 to generate a set of fan beam projections P.sub.f (.beta.) at a corresponding set of projection angles .beta.. In a conventional scan, the patient remains at a constant Z-axis position during the scan, whereas in a volumetric (and more specifically described herein as "helical") CT scan, the patient and rotating disk are translated relative to one another along the Z-axis while the disk is rotated about the patient. FIG. 3A schematically illustrates the volume within which the data is collected during a conventional scan, and FIG. 3B illustrates the volume within which the data is collected during a helical scan. As shown in FIG. 3A, if the X-ray source 12 and detector system 14 are rotated about an object 20 while the object 20 remains at a fixed Z-axis location, the rays associated with all the projections collected by detector system 14 will all lie in a common "slice plane" 50. As shown in FIG. 3B, if the object 20 and rotating disk are continuously translated relative to one another in the direction of the Z-axis while the disk is rotated about the object 20, none of the scanning planes associated with the corresponding projections will lie in a common slice plane. Rather, the scanning plane associated with each projection will lie at a unique position along the Z-axis. FIG. 3B illustrates the Z-axis coordinate of the scanning planes corresponding to helical projection angles in the interval (0,10.pi.), i.e., five complete rotations of the disk. Since the data values of each projection depend on the Z-axis location of the patient, each projection may be considered as a function of two variables, .beta. and z, where .beta. represents projection angle and where z represents the Z-axis coordinate of the scanning plane associated with that projection.
Since the patient remains at a constant Z-axis position during a conventional scan, this type of scanning is commonly referred to as "Constant Z-axis position scanning" or CZA scanning. In helical scanning, the projections P.sub.f (.beta.,z) are normally acquired such that the Z-axis coordinate z is linearly related to the projection angle .beta. so that z (.beta.)=c.beta., where c is a constant. This form of helical scanning is often referred to as Constant Speed Helical (CSH) scanning.
Using well known algorithms, such as the inverse Radon transform, a tomogram may be generated from a set of projections that all share the same scanning plane, and this common scanning plane is referred to as the "slice plane". A tomogram is representative of the density of a two dimensional "slice", along the slice plane, of the object being scanned. The process of generating a tomogram from the projections is commonly referred to as "filtered back projection" or "reconstruction", since the tomogram may be thought of as being reconstructed from the projection data. The signal processing portion of a CT scanner normally includes a back projector for generating the tomograms from the projections.
In CZA scanning, all the projections share a common scanning plane, so these projections may be applied directly to the back projector for generation of a tomogram. In CSH scanning, each projection has a unique scanning plane located at a unique Z-axis coordinate, so CSH projections may not be applied directly to a back projector. However, as is well known, the data collected during a CSH scan may be interpolated in various fashions to generate a set of interpolated projections that do all share a common scanning plane. Each interpolated projection may be generated by combining two projections taken at equivalent projection angles and at different Z-axis positions. These interpolated projections may be treated as CZA data and may be applied to a back projector to generate a tomogram.
Thus, CSH scanning requires some form of interpolation to generate a tomogram, which is disadvantageous since data interpolated from different Z-axis positions tends to contribute to image artifacts because of inconsistency of the projections from different z positions. Also, since the CSH projection data, which is collected over an interval of Z-axis locations, is combined to generate the interpolated CZA data, tomograms generated during CSH scanning have a wider effective slice plane width than tomograms generated by CZA scanning. However, CSH scanning advantageously permits rapid scanning of a large volume of a patient. For example, in a time interval short enough for a patient to comfortably hold his/her breath (and thereby remain relatively motionless), a CSH scan may collect enough data to fully scan an entire organ such as a kidney. In practice, the disk may, for example, be rotated on the order of 40.pi. radians (i.e., 20 complete revolutions) during a single helical scan, and the data collected may be used to generate tomograms at a plurality of slice planes.
Fan beam projection data may be applied directly to a back projector that uses a fan beam reconstruction algorithm. However, as is well known, fan beam projection data is often reordered and interleaved to generate parallel beam projection data, and this parallel beam data may be applied to a back projector that uses a parallel beam reconstruction algorithm. FIG. 4A illustrates some of the individual rays used to generate a fan beam projection 100 taken at a fan beam projection angle of zero degrees, and FIG. 4B illustrates some of the individual rays used to generate a parallel beam projection 102 taken at a parallel beam projection angle of zero degrees. As shown, none of the rays in the fan beam projection 100 are parallel to one another, while all of the rays in the parallel beam projection 102 are mutually parallel. Since all of the rays emanate from the focal spot of the X-ray source 12 to form a fan beam, the CT scanner does not generate all the rays of a parallel beam projection simultaneously. However as is well known, the fan beam projection data may be reordered and interleaved to generate parallel beam projections.
FIGS. 5A and 5B illustrate a method of generating a reordered projection. FIGS. 5A and 5B show the positions of X-ray source 12 and detector system 14 during generation of two successive fan beam projections. For ease of illustration, FIGS. 5A and 5B show detector system 14 as including seven individual detectors, 22:1, 22:2, 22:3, 22:4, 22:5, 22:6, and 22:7, although it is well understood that detector systems typically have detectors numbering in the hundreds. As shown, during a scan, X-ray source 12 and detector system 14 rotate in a counter clockwise direction about the Z-axis. During the first fan beam projection, shown in FIG. 5A, a ray 114 is incident on a detector 22:4 (i.e., the detector in the fourth channel of detector system 14). During the next fan beam projection, shown in FIG. 5B, a ray 116 is incident on detector 22:3 (i.e., the detector in the third channel of detector system 14). When the spacing between the individual detectors is matched to the amount of disk rotation between generation of successive fan beam projections, the ray 114 is parallel to, and slightly offset from, ray 116. When this basic relationship is true for all detectors and all fan beam projections, any two rays incident on adjacent detectors during successive fan beam projections are parallel and are slightly offset from each other. In the Anatom scanner, which is manufactured by the assignee of the present invention, the detector system includes three hundred eighty four individual detectors, each of the individual detectors being spaced apart by a detector angle of 0.125.degree.. Consequently, in the Anatom scanner successive fan beam projections are separated by a projection angle that is also equal to 0.125.degree.. This allows the fan beam data collected by that scanner to be reordered into reordered projections, such that the rays used to generate the data points in each reordered projection are all mutually parallel.
Each reordered projection P.sub.r (.beta.) has an associated reordered projection angle .beta. and may be represented as a set of data points R.sub.1, R.sub.2, R.sub.3, . . . , R.sub.N. Each reordered projection includes the same number of data points as a fan beam projection. The reordered projection angle .beta. is the angle between the rays used to generate the data points of the reordered projection and the vertical axis. No two data points in a single reordered projection are generated at the same fan beam projection angle. Rather, each pair of adjacent data points in a reordered projection are generated by adjacent detectors at adjacent fan beam projection angles. So when CSH projection data is reordered into reordered projections, no two data points in a single reordered projection are generated at a common Z-axis position. However, since all the data points in a reordered projection are generated by a set of adjacent fan beam projections, the data points in a reordered projection are generated at Z-axis positions that are relatively proximal to one another.
Each reordered projection P.sub.r (.beta.) has an associated reordered projection P.sub.r (.beta.+.pi.) generated 180.degree. away. These two associated reordered projections are often combined, or interleaved, to generate a single parallel beam projection at a parallel beam projection angle of .beta..
In a detector system that is both symmetric and centered, the rays used to generate the reordered projection P.sub.r (.beta.) are coincident and antiparallel ("antiparallel" because the relative positions of the detectors and the X-ray source are reversed) with the rays used to generate the associated reordered projection P.sub.r (.beta.+.pi.). So in the absence of patient motion, the data in the set of reordered projections with .beta. in the interval (0,.pi.) is identical to the data in the set of reordered projections with .beta. in the interval (.pi.,2.pi.). As is well known, detector offsets are often used to avoid this redundancy and to thereby increase the number of unique sampling points measured with every single complete rotation of the disk. In a detector system that has a detector offset, the rays used to generate the reordered projection P.sub.r (.beta.) are parallel to and slightly offset from the rays used to generate the reordered projection P.sub.r (.beta.+.pi.).
FIGS. 6A and 6B illustrate the spatial relationship between X-ray source 12, a cross section of patient 20, and detector system 14 for projection angles of zero and 180 degrees, respectively. The illustrated detector system 14 is characterized by a quarter detector offset, meaning that the center line 40 intersects the central detector 22:4 at a point that is offset from the center of that detector by one quarter of the detector's width. This detector offset insures that the detector system 14 at 0.degree. is offset from the detector system 14 at 180.degree. by a half detector spacing. In FIGS. 6A and 6B, detector system 14 is once again shown for convenience of illustration as including seven individual detectors.
FIG. 7 illustrates two reordered projections 110, 112 generated from data collected with the quarter detector offset detector system 14, such as shown in FIGS. 6A and 6B, except for N detectors. FIG. 7 also shows a parallel beam projection 116 generated by combining the data in reordered projections 110, 112. In FIG. 7, the X-axis represents the relative linear spatial position of the ray used to generate each data point (i.e., the ray used to generate the data point R.sub.3 in projection 110 is to the left of the ray used to generate the data point R.sub.4 in projection 110). Reordered projections 110 and 112 are generated at reordered projection angles of 0.degree. and 180.degree., respectively. Due to the quarter detector offset in detector system 14, the X-axis coordinate of the data point R.sub.i in reordered projection 112 is closer to the X-axis coordinates of the data points R.sub.i and R.sub.i+1 in the reordered projection 110 than to any other data points in either of projections 110, 112. Although reordered projections 110, 112 are generated 180.degree. apart from one another, the X-axis coordinates of each data point in projection 110 is closer to the coordinates of two data points in projection 112 than to the X-axis coordinates of any other data point in projection 110. This relationship allows any two reordered projections generated 180.degree. apart to be interleaved to generate a single parallel beam projection.
Parallel beam projection 116 is formed by combining or interleaving the data in reordered projections 110, 112. Parallel beam projection 116 includes a set of data points P.sub.i for all integers i from one to 2N. So parallel beam projection 116 includes twice as many data points as any of the single reordered projections 110, 112. In parallel beam projection 116, the odd data point P.sub.2i+1 equals the data point R.sub.i+1 in reordered projection 110, and the even data point P.sub.2i+2 equals the data point R.sub.i+1 in reordered projection 112 for all integers i from zero to N minus 1. So in parallel beam projection 116, all of the odd data points (e.g., P.sub.1 and P.sub.3) are contributed by reordered projection 110 and all of the even data points (e.g., P.sub.2 and P.sub.4) are contributed by reordered projection 112.
FIGS. 8A, 8B, and 8C show graphs that illustrate the relationship between the fan beam projection angle and the parallel beam projection angle of data points in interleaved parallel beam projections generated by a symmetric offset detector array. More specifically, these figures illustrate how fan beam projections for fan beam projection angles in the interval (0, 2.pi.) may be reordered and interleaved to generate a set of interleaved parallel beam projections for parallel beam projection angles in the interval (-.pi./2, .pi./2). As is well known, back projectors using parallel beam filtered back projection algorithms may generate tomograms from parallel beam projections with parallel beam projection angles in the interval (-.pi./2, .pi./2).
FIG. 8A shows a graph that illustrates the relationship between reordered and fan beam projection angles. In FIG. 8A, the Y-axis represents reordered projection angle and the X-axis represents fan beam projection angle. The two lines indicated at A represent the data collected by detector A (as shown in FIG. 2), which is located at a detector angle of -.gamma..sub.m ; the two lines indicated at B represent data collected by detector B, which is located at a detector angle of .gamma..sub.m ; and the two lines indicated at C represent data collected by the central detector C, which is located at a detector angle of 0.degree.. Since detectors A and B are located at opposite ends of the detector system, any horizontal line extending from line A to line B represents the data in a single reordered projection. Similarly, any vertical line extending from line A to line B represents the data in a single fan beam projection. For example, horizontal line 130 represents the data in a reordered projection for a reordered projection angle of .pi./2. The reordered projection indicated by line 130 is formed by data collected at fan beam projection angles in the continuous interval (.pi./2-.gamma..sub.m, .pi./2+.gamma..sub.m) equal to the angle subtended by the fan beam between the lines A and B. Most reordered projections may similarly be formed from data collected over a continuous interval of fan beam projections. However, all the reordered projections for reordered projection angles in the range (-.gamma..sub.m, .gamma..sub.m) are formed from data collected over two non-adjacent intervals of fan beam projection angles. For example, a reordered projection for a reordered projection angle of 0.degree. is formed by data collected by the detectors between detectors C and B over fan beam projection angles in the interval (0, .gamma..sub.m), and also from data collected by the detectors between detectors A and C over fan beam projection angles in the interval (2.pi.-.gamma..sub.m, 2.pi.). So the reordered projections for reordered projection angles in the range (-.gamma..sub.m, .gamma..sub.m) are formed from some data collected at the beginning of a disk rotation (i.e., near fan beam projection angle of 0.degree.), and from some data collected near the end of a disk rotation (i.e., near fan beam projection angle of 2.pi.).
FIG. 8B shows a graph that is equivalent to the graph shown in FIG. 8A, however, in FIG. 8B, the Y-axis represents parallel beam projection angles rather than reordered projection angles. Since each parallel beam projection is formed by combining data from two reordered projections, the parallel beam projection angles represented by the Y-axis in FIG. 8B only extend over the interval (-.pi./2, .pi./2) rather than over the interval (-.pi., .pi.). In FIG. 8B, the data collected by detector A over fan beam projection angles in the intervals (0, .pi./2-.gamma..sub.m), (.pi./2-.gamma..sub.m, 3.pi./2-.gamma..sub.m), and (3.pi./2-.gamma..sub.m, 2.pi.) are represented by lines A', A, and A", respectively. The data collected by detector C over fan beam projection angles in the intervals (0, .pi./2), (.pi./2, 3.pi./2), and (3.pi./2, 2.pi.) are represented by lines C', C, and C", respectively. The data collected by detector B over fan beam projection angles in the intervals (0, .pi./2+.gamma..sub.m), (.pi./2+.gamma..sub.m, 3.pi./2+.gamma..sub.m), and (3.pi./2+.gamma..sub.m, 2.pi.) are represented by lines B', B, and B", respectively.
FIG. 8C shows a graph that indicates how the data shown in FIG. 8B may be interleaved to generate interleaved parallel beam projections. The data indicated at lines B' and B" are interleaved with the data indicated at line A, the data indicated at lines C' and C" are interleaved with the data indicated at line C, and the data indicated at lines A' and A" are interleaved with the data indicated at line B. So when forming parallel beam projections, the data from the central detector C is interleaved with other data collected by the central detector (as indicated by C' and C"). However, the central detector is the only detector in the system to have this property. Each remaining detector is interleaved with a detector on the opposite side of the detector system to generate the parallel beam projections. For example, the data collected by detector A is interleaved with data collected by detector B to form a parallel beam projection. In FIG. 8C, interleaved parallel beam projections are represented by horizontal lines. For example, line 132 represents the data in the parallel beam projection at the parallel beam projection angle of .pi./4.
In an interleaved parallel beam projection, the ray path used to generate the ith data point P.sub.i is closer to the ray paths used to generate the adjacent data points P.sub.i-1 and P.sub.i+1 than to any other ray paths. However, the difference between the measurement times of adjacent data points (e.g., P.sub.i and P.sub.i-1) is much greater than the difference between the measurement times of alternate data points (e.g., P.sub.i and P.sub.i-2). For example, if T.sub.i represents the time that a data point P.sub.i is measured, then T.sub.i minus T.sub.i-1 is much greater than T.sub.i minus T.sub.i-2. This is true because all of the even points of a single parallel beam projection are contributed by a single reordered projection (and all the data points of a reordered projection are generated by a set of adjacent fan beam projections). However, adjacent data points in the parallel beam projection are contributed by two different reordered projections generated 180.degree. apart from one another. So the measurement times of such adjacent data points are separated by the time required for the disk to rotate approximately 180.degree..
In the absence of patient motion (i.e., in a CZA scan during which the patient does not move) the portions of the patient measured by adjacent data points in a parallel beam projection are proximal to one another. However, in a parallel beam projection generated from CSH data, the portions of the patient measured by adjacent data points are axially separated by a relatively large distance because the relative movement of the patient and the rotating disk is a translation of a considerable distance during the time required for the disk to rotate approximately 180.degree.. This leads to a discrepancy between the even data points and the odd data points in every single parallel beam projection generated during a CSH scan. These discrepancies appear as high frequency noise in the projection data and complicate the process of generating tomograms from CSH data.
FIGS. 9A and 9B illustrate one prior art method for generating a tomogram from CSH data generated by a symmetric centered detector system. FIG. 9A shows a graph of some of the data points in a single interleaved parallel beam projection generated by a symmetric centered detector system during a CSH scan at a parallel beam projection angle of .beta.. In FIG. 9A, the Y-axis represents the amplitude of the data points of the projection, and the X-axis represents the relative linear spatial position of the rays used to generate the data points of the projection. Since the illustrated projection was generated using a centered detector system, the rays used to generate the data points P.sub.1 and P.sub.2 (which are 180.degree. apart) are coincident, so these data points share the same X-axis coordinate. Since the illustrated projection was generated during a CSH scan, there tends to be a discrepancy between the amplitudes of the odd data points and the even data points. As was stated above, this discrepancy occurs because the odd data points are generated at Z-axis locations that are displaced from the Z-axis locations where the even data points are generated.
In a typical CSH scan, the collected data is used to generate a tomogram at a slice plane having a Z-axis location z.sub.sp. Although the odd and even data points in each parallel beam projection are all generated at slightly different Z-axis locations, all of the odd data points are typically generated on one side of the slice plane, and similarly, all of the even data points are generated on the other side of the slice plane, i.e., the slice plane is positioned along the Z-axis between the two sets of odd and even data points. Interpolation is normally performed between each pair of corresponding data points displaced 180.degree. to estimate the value of a projection at the location of the slice plane z.sub.sp. FIG. 9B illustrates the results of this type of interpolation. As shown, the value of an interpolated data point P'.sub.1 at the slice plane is generated by averaging the data points P.sub.1 and P.sub.2, the value of an interpolated data point P'.sub.3 at the slice plane is generated by averaging the data points P.sub.3 and P.sub.4, and so on. The interpolated data points, which are all illustrated as dashed-circles, represent a non-interleaved parallel beam projection at the slice plane z.sub.sp for a projection angle of .beta.. Similar interpolations are performed for every pair of parallel beam projections at projection angles 180.degree. apart for each parallel beam projection angle .beta. and the associated .beta.+180.degree., and then the interpolated parallel beam projections are filtered and applied to a back projector for generation of a tomogram. It is not necessary to perform further interpolation on the parallel beam projections. The interpolation coefficient for each data point at a projection angle can be treated as a weighting factor for that data point, and the weighting factor can be applied to the original data point in the fan beam projection. In this way, the interpolation to the slice plane z.sub.sp is performed up-front in the fan beam projection by weighting each data point according to its Z-axis position. It is not necessary to reorder the data into parallel beam projections either. The tomogram can be generated from the weighted fan beam projections by using a fan beam filtered backprojection.
FIG. 10A shows an example of a graph of some of the data points in a single interleaved parallel beam projection generated by a symmetric offset detector system during a CSH scan at a parallel beam projection angle of .beta.. Since the detector system is characterized by an offset, each data point in the projection has a unique X-axis coordinate position, and the odd data points are interleaved with the even data points along the X-axis. Prior art methods of generating tomograms from such data normally involve performing an interpolation by weighting the original fan beam projection data similar to the methods discussed above in connection with FIGS. 9A and 9B. Each original data point is multiplied by a weighting factor and thereby shift the amplitude values of the odd and even data points towards each other as illustrated in FIG. 10B. These weighted original data points may then be used to generate the tomogram by a fan beam filtered backprojection. During the backprojection, the weighted data from fan beam projection at 180.degree. apart are superimposed such that it is almost equivalent to having data interpolated to the slice plane position. The weighted original data can also be reordered into parallel beam projections for a parallel beam filtered backprojection. In this method, the filtering process of convolution performs some degree of averaging for adjacent even and odd points such that the weighting in the original data is almost equivalent to having even and odd data points in the interleaved parallel beam projection interpolated to the slice plane.
In every pair of corresponding original data points (e.g., P.sub.1 and P.sub.2), one of the data points is generated at a first set of X and Z-axis coordinates (x.sub.1, z.sub.1) and the other data point is generated at a second set of X and Z-axis coordinates (x.sub.2, z.sub.2). The weighting factors are selected so as to generate an equivalent interpolated data point having a Z-axis coordinate at the slice plane z.sub.sp. However, these prior methods do not fully deal with the fact that the two original data points are generated at different X-axis coordinates. The weighting factor w ranges from 0.0 to 1.0. When the weighted data are reordered into an interleaved parallel beam projection, the projection amplitudes between adjacent data points can be huge because the even and odd data points have been multiplied by w and 1-w, respectively. The normal filter for convolution and the backprojection can reduce, but not completely remove, the amplitude differences between the even and odd data points. Consequently, the tomogram may contain a relatively large amount of artifacts due to the large amplitude differences between even and odd data points. It is possible to remove these artifacts by using a filter with a lower frequency response. But, by doing so, the spatial resolution of the tomogram is reduced. By the time these artifacts are fully removed, the spatial resolution is degraded to the level as if the data were collected and processed without the offset in the detector system. In the same situation, if the tomogram is generated from the weighted data by fan beam filtered backprojection method, the spatial resolution must be reduced to avoid these weighting induced artifacts. Again, the higher resolution advantage of an offset detector system will be lost by the time these artifacts are completely removed. There is therefore a need for an improved method of processing the data collected by an offset detector system for a CSH scan.
Another problem with prior art helical scanning techniques relates to use of asymmetric detector systems. FIG. 11 illustrates the geometry of a CT scanner having an asymmetric detector system 14. This detector system includes a symmetric portion 14a extending from detector angle -.gamma..sub.m +.alpha. to .gamma..sub.m -.alpha., and an asymmetric portion 14b extending from detector angle .gamma..sub.m -.alpha. to .gamma..sub.m, where .alpha. is the angular extent of the asymmetric portion. Detector system 14 may also be thought of as not including a portion 14c extending from detector angle -.gamma..sub.m to -.gamma..sub.m +.alpha.. If detector system 14 did include the missing portion 14c, then the detector system would be symmetric.
Such asymmetric detector systems are often used in CT scanners so as to increase the "field of view" (FOV) of the scanner without significantly increasing the cost of the detector system and associated DAS. The FOV of a scanner is determined by the effective angular extent of the detector system. For example, without rotation the angular extent of a scanner (i.e., the angle subtended by the fan beam) using the symmetric detector system illustrated in FIG. 2 is equal to 2.gamma..sub.m, and the angular extent of a scanner (i.e., the angle subtended by the fan beam) using the asymmetric detector system illustrated in FIG. 11 is equal to 2.gamma..sub.m -.alpha.. But with rotation of the detector system over 360 degrees during a scan, both give effective angular extent of 2.gamma..sub.m. So use of the asymmetric detector system achieves the same FOV as the symmetric detector system and saves the detectors over the angle .alpha.. As is well known, the importance of each detector (in terms of its contribution to tomograms) decreases with increasing detector angle. So it is reasonable to eliminate half the detectors beyond a certain detector angle and thereby generate an asymmetric detector system. By way of example, the above-referenced Anatom scanner uses an asymmetric detector system where .gamma..sub.m is equal to 28.85.degree., and .alpha. is equal to 9.69.degree.. Although such asymmetric detector system are popular, their use complicates the process of generating helical scans.
FIG. 12 illustrates the X-axis coordinates of the data points in two reordered projections 210, 212, and also of two parallel beam projections 214, 216. Projections 210, 212, 214, and 216 are generated from data collected by an asymmetric offset detector system 14 (as shown in FIG. 11) that includes N individual detectors. The asymmetric portion 14b of detector system 14 is assumed to include j individual detectors, and the symmetric portion 14a therefore includes N-j individual detectors.
Each of the reordered projections 210, 212 includes N data point R.sub.k for all integers k from one to N. Reordered projection 210 is generated at a reordered projection angle of .beta., and reordered projection 212 is generated at a reordered projection angle of .beta.+.pi., so these projections may be combined to generate a parallel beam projection. In reordered projection 210, the data points R.sub.1 through R.sub.j are generated by the asymmetric portion 14b, and the data points R.sub.j+1 through R.sub.N are generated by the symmetric portion 14a. In reordered projection 212, the data points R.sub.1 through R.sub.N-j are generated by the symmetric portion 14a, and the data points R.sub.N-j+1 through R.sub.N are generated by the asymmetric portion 14b. The data points collected by the symmetric portion 14a of the detector system (i.e., data points R.sub.j+1 to R.sub.N in projection 210, and data points R.sub.1 to R.sub.N-j in projection 212) may be interleaved to generate a parallel beam projection. However, as illustrated in FIG. 11, no data points are available for interleaving the data points collected by the asymmetric portion 14b. Such data points could only have been collected by the missing portion 14c of the detector system 14. It is this inability to interleave the data collected by the asymmetric portion 14b that complicates the process of CSH scanning with an asymmetric detector system.
Parallel beam projection 214 is generated by combining the reordered projections 210, 212. Since no data points are available for interleaving the data collected by the asymmetric portion 14b of the detector system 14, parallel beam projection 214 includes a central interleaved portion and two exterior non-interleaved portions. More specifically, parallel beam projection 214 includes an exterior region composed of data points P.sub.1 to P.sub.j, and these data points are equal to the data points R.sub.1 to R.sub.j, respectively, of reordered projection 210. Parallel beam projection 214 also includes an exterior region composed of data points P.sub.2N-j+1 to P.sub.2N, and these data points are equal to the data points R.sub.N-j+1 to R.sub.N, respectively, of reordered projection 212. Finally, parallel beam projection 214 includes a central interleaved portion composed of data points P.sub.j+1 to P.sub.2N-j, and these data points are generated by combining the data points R.sub.j+1 to R.sub.N from reordered projection 210 and the data points R.sub.1 to R.sub.N-j from reordered projection 212. More specifically, the data point P.sub.j+1 equals the data point R.sub.j+1 from projection 210, the data point P.sub.j+2 equals the data point R.sub.1 from projection 212, the data point P.sub.j+3 equals the data point R.sub.j+2 from projection 210, the data point P.sub.j+4 equals the data point R.sub.2 from projection 212, and so on. Since the data points in the exterior regions of the parallel beam projection are not interleaved, the spacing between adjacent data points in these regions is double the spacing between adjacent data points in the central region.
Parallel beam projection 216 includes exactly the same data as parallel beam projection 214. However, the indices in parallel beam projection 216 have been altered to reflect the difference in the spacing between data points in the exterior regions and in the central region. More specifically, in parallel beam projection 216, the indices of the data points have been altered so as to eliminate even data points from one of the exterior (non-interleaved) regions and to eliminate odd data points from the other exterior region. The odd data points P.sub.1, P.sub.3, . . . , P.sub.2j-1, form one of the exterior regions of projection 216, and these data points are of course equal to the data points P.sub.1 to P.sub.j, respectively, of projection 214. Similarly, the even data points P.sub.2N+2, P.sub.2N+4, . . . , P.sub.2N+2j, form the other exterior region of projection 216, and these data points are equal the data points P.sub.2N-j+1 to P.sub.2N, respectively, of projection 214. Finally, the even and odd data points P.sub.2j+1 to P.sub.2N form the central (interleaved) region of projection 216, and these data points are equal to the data points P.sub.j+1 to P.sub.2N-j, respectively, of projection 214. The representation for the data points used in parallel beam projection 216 is convenient since in this projection, all the odd data points are contributed by one of the reordered projections (i.e., projection 210) and all the even data points are contributed by the other reordered projection (i.e., projection 212). Further, in projection 216, the spacing between any two data points can be calculated as a simple function of the indices of the data points.
Prior art helical scanning techniques do not work well when applied to parallel beam projections collected by an asymmetric detector system such as projection 216. Interpolation may be performed on the interleaved data points in the central region of such projections to generate data points at the slice plane. However, since the data in the exterior regions is not interleaved, this data may not be interpolated to generate data points at the slice plane. There is therefore a need for a CSH scanning technique that generates tomograms from projections collected by asymmetric offset detector systems.